Tykhonov well-posedness of split problems

Qiao-Yuan Shu,Mircea Sofonea,Yi-Bin Xiao

doi:10.1186/s13660-020-02421-w

Abstract

In (J. Optim. Theory Appl. 183:139–157, 2019) we introduced and studied the concept of well-posedness in the sense of Tykhonov for abstract problems formulated on metric spaces. Our aim of this current paper is to extend the results in (J. Optim. Theory Appl. 183:139–157, 2019) to a system which consists of two independent problems denoted by P and Q, coupled by a nonlinear equation. Following the terminology used in literature we refer to such a system as a split problem. We introduce the concept of well-posedness for the abstract split problem and provide its characterization in terms of metric properties for a family of approximating sets and in terms of the well-posedness for the problems P and Q, as well. Then we illustrate the applicability of our results in the study of three relevant particular cases: a split variational–hemivariational inequality, an elliptic variational inequality and a history-dependent variational inequality. We describe each split problem and clearly indicate the family of approximating sets. We provide necessary and sufficient conditions which guarantee the well-posedness of the split variational–hemivariational inequality. Moreover, under appropriate assumptions on the data, we prove the well-posedness of the split elliptic variational inequality as well as the well-posedness of the split history-dependent variational inequality. We illustrate our abstract results with various examples, part of them arising in contact mechanics.

Highlights

1 Introduction Variational inequalities and their related problems represent a powerful mathematical tool used in the study of various nonlinear boundary value problems
Split variational inequalities have many applications in practical problems arising from signal recovery, image processing and radiation therapy
We introduce the concept of split problem and its wellposedness

Summary

Introduction

Variational inequalities and their related problems represent a powerful mathematical tool used in the study of various nonlinear boundary value problems. We introduce the concept of split problem and its wellposedness To this end, besides Problem P endowed with its set of solutions SP and the associated family of approximating sets {ΩP(ε)}ε>0 which satisfy condition (2.3), we consider a second problem, denoted by Q, defined on the space Y. We say in what follows that M represents a split problem Note that this new mathematical object is constructed by using the problems P and Q, together with their sets of solutions SP and SQ, the families of approximating sets {ΩP(ε)}ε>0 and {ΩQ(ε)}ε>0, the operator G and the element f. Consider the split problem M in Example 9 Note that in this case diam(ΩM(ε)) ≥ x – x R2 ≥ 2 for each ε > 0 and, Theorem 10 implies that this problem is not well-posed. This result recovers one of the implication of the Claim 1 in Example 8

A split variational–hemivariational inequality

A well-posed split variational inequality